State Biot-Savart's law

Statement. The magnetic field intensity dH produced at a point P by a differential current element I dL is directly proportional to the product I dL and the sine of the angle between the element and the line joining it to P, and inversely proportional to the square of the distance R between them.

dH = I dL × a_R / (4πR²)       (A/m)
dH = I dL sinθ / (4πR²)        (magnitude)

Here R is the distance from the element to P, a_R is the unit vector from the element to P, and θ is the angle between dL and a_R.

        I dL
   →→→→→●→→→→→
         \  θ
          \ R
           \
            ● P     dH is perpendicular to the plane of dL and R
                    (into the page in the figure; right-hand rule)

Explanation

• The direction of dH is that of dL × a_R, given by the right-hand rule: point the right thumb along the current, and the fingers curl in the direction of H.
• For a complete circuit, H = ∮ I dL × a_R/(4πR²).
• For surface or volume currents: H = ∫ K × a_R dS/(4πR²) or ∫ J × a_R dv/(4πR²).
• The law is the magnetic counterpart of Coulomb's law, since the field falls as 1/R². An isolated current element cannot exist, but the law is valid for closed circuits.

Example. For an infinite straight wire, integrating the law gives H = I/(2πρ) a_φ.